For many examples of wordhyperbolic groups which I have seen in the context of lowdimensional topology, the ideal boundary is either homeomorphic to a nsphere for some n or a Cantor set. So, I was wondering if this is generally true or there are some examples of hyperbolic groups whose boundaries are neither a sphere of some dimension nor a Cantor set. If there exist such examples, is there any known topological classification of boundaries of hyperbolic groups?

$\begingroup$ The class of discrete hyperbolic groups is stable under free products, this provides lowcost examples with a different boundary. For more refined ones, see this survey by I. Kapovich and Benakli: math.uiuc.edu/~kapovich/PAPERS/bry1.pdf $\endgroup$ – YCor Jul 4 '13 at 21:33
There are plenty of other possibilities. Here are a few examples:
The boundary of the fundamental group of an acylindrical hyperbolic 3manifold with totally geodesic boundary is homeomorphic to a Sierpinski carpet. (This appears in the Kapovich and Kleiner paper mentioned below, but must be standardthe point is the topological fact that any planar continuum with no local cut points is homeomorphic to a Sierpinski carpet. Indeed, they prove a converse result, modulo the Cannon Conjecture.)
The boundary of a random group is homeomorphic to a Menger sponge (DahmaniGuirardelPrzytycki).
Bowditch proved that cyclic splittings of $\Gamma$ correspond to local cut points on the boundary. In particular, the boundary of a graph of free groups with cyclic edge groups can be decomposed along cut pairs into Cantor sets.
The classification of 1dimensional Polish spaces implies that this is a complete list of boundaries of 2dimensional hyperbolic groups, in the sense that if the boundary is connected with no local cut point (ie the group has no splitting over a virtually cyclic subgroup) then it must be a Sierpinski carpet or Menger sponge (KapovichKleiner).
I imagine that a classification in higher dimensions is completely out of reach, though I'm no expert. (Misha Kapovich is active on MO and can provide an authoritative answer.)
I'll add full references tomorrow.

$\begingroup$ Thanks a lot, Henry. This is very interesting. I wonder if there is any partial result in classification of boundaries of 3dimensional hyperbolic groups. $\endgroup$ – Harry Baik Jul 5 '13 at 2:36

$\begingroup$ @harry: what do you mean by "3dimensional hyperbolic group"? there are many inequivalent notions of dimension. $\endgroup$ – YCor Jul 5 '13 at 6:42

$\begingroup$ @Yves, in this case, the natural definition of dimension for a hyperbolic group $\Gamma$ is the topological dimension of the boundary +1. As you say, there are other relevant definitions. It's unknown which hyperbolic groups have boundary of conformal dimension 1, for instance. $\endgroup$ – HJRW Jul 5 '13 at 9:02

$\begingroup$ @YvesCornulier: Yes, I just meant the topological dimension of the boundary + 1. $\endgroup$ – Harry Baik Jul 6 '13 at 2:37

$\begingroup$ @HJRW I think you mean Sierpinski carpet as opposed to Sierpinski gasket. (The latter term is a synonym for the Sierpinski triangle.) $\endgroup$ – Jim Belk Jul 22 '18 at 13:22